{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "from d2l import torch as d2l\n",
    "import torch\n",
    "import torch.nn as nn"
   ]
  },
  {
   "source": [
    "## 8.1.2。训练\n",
    "在回顾了许多统计工具之后，让我们在实践中尝试一下。我们首先生成一些数据。为了简单起见，我们使用正弦函数并在时间步长中添加一些附加噪声来生成序列数据 1,2,…,1000 ."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
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     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "T = 1000  # Generate a total of 1000 points\n",
    "time = torch.arange(1, T + 1, dtype=torch.float32) #1,2,...,1000\n",
    "x = torch.sin(0.01 * time) + torch.normal(0, 0.2, (T,))\n",
    "d2l.plot(time, [x], 'time', 'x', xlim=[1, 1000], figsize=(6, 3))"
   ]
  },
  {
   "source": [
    "接下来，我们需要将这样的序列转换为模型可以训练的特征和标签。根据嵌入尺寸 $\\tau$ 我们将数据成对映射 $y_t = x_t$ 和 $\\mathbf{x}_t = [x_{t-\\tau}, \\ldots, x_{t-1}]$.\n",
    "精明的读者可能已经注意到，这给了我们 $\\tau$  数据示例较少，因为第一个示例没有足够的历史记录  $\\tau$ 其中。一个简单的解决方法，尤其是如果序列很长的话，就是丢弃这几个术语。或者，我们可以用零填充序列。在这里，我们仅使用前600个特征标签对进行训练"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "tensor([[0., 0., 0., 0.],\n        [0., 0., 0., 0.],\n        [0., 0., 0., 0.]])"
     },
     "metadata": {},
     "execution_count": 56
    }
   ],
   "source": [
    "tau = 4\n",
    "features = torch.zeros((T - tau, tau)) # torch.Size([996, 4])\n",
    "features[0:3,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "tensor([[1., 2., 3., 4.],\n        [2., 3., 4., 5.],\n        [3., 4., 5., 6.]])"
     },
     "metadata": {},
     "execution_count": 57
    }
   ],
   "source": [
    "features2 = torch.zeros((T - tau, tau)) # torch.Size([996, 4])\n",
    "for i in range(tau):\n",
    "    features2[:, i] = time[i: T - tau + i]\n",
    "features2[0:3,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "tensor([[5.],\n        [6.],\n        [7.]])"
     },
     "metadata": {},
     "execution_count": 58
    }
   ],
   "source": [
    "labels2 = d2l.reshape(time[tau:], (-1, 1)) # torch.Size([996, 1])\n",
    "labels2[0:3,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": "X2 tensor([[198., 199., 200., 201.],\n        [139., 140., 141., 142.],\n        [  4.,   5.,   6.,   7.],\n        [323., 324., 325., 326.],\n        [249., 250., 251., 252.],\n        [202., 203., 204., 205.]]) \ny2: tensor([[202.],\n        [143.],\n        [  8.],\n        [327.],\n        [253.],\n        [206.]])\n"
    }
   ],
   "source": [
    "batch_size2, n_train2 = 6, 600\n",
    "# Only the first `n_train` examples are used for training\n",
    "train_iter2 = d2l.load_array((features2[:n_train2], labels2[:n_train2]),\n",
    "                            batch_size2, is_train=True)\n",
    "for X2, y2 in train_iter2:\n",
    "    print('X2',X2,'\\ny2:',y2)\n",
    "    break"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "tensor([[ 0.0093, -0.2983, -0.1994, -0.1204],\n        [-0.2983, -0.1994, -0.1204, -0.1515],\n        [-0.1994, -0.1204, -0.1515,  0.2984]])"
     },
     "metadata": {},
     "execution_count": 60
    }
   ],
   "source": [
    "for i in range(tau):\n",
    "    features[:, i] = x[i: T - tau + i]\n",
    "features[0:3,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "tensor([[-0.1515],\n        [ 0.2984],\n        [ 0.2316]])"
     },
     "metadata": {},
     "execution_count": 61
    }
   ],
   "source": [
    "labels = d2l.reshape(x[tau:], (-1, 1)) # torch.Size([996, 1])\n",
    "labels[0:3,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "text": "\u001b[1;31mSignature:\u001b[0m \u001b[0md2l\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mload_array\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdata_arrays\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mbatch_size\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mis_train\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;32mTrue\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;31mSource:\u001b[0m   \n\u001b[1;32mdef\u001b[0m \u001b[0mload_array\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdata_arrays\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mbatch_size\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mis_train\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;32mTrue\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m  \u001b[1;31m#@save\u001b[0m\u001b[1;33m\n\u001b[0m    \u001b[1;34m\"\"\"Construct a PyTorch data iterator.\"\"\"\u001b[0m\u001b[1;33m\n\u001b[0m    \u001b[0mdataset\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdata\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mTensorDataset\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0mdata_arrays\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\n\u001b[0m    \u001b[1;32mreturn\u001b[0m \u001b[0mdata\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mDataLoader\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdataset\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mbatch_size\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mshuffle\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mis_train\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;31mFile:\u001b[0m      d:\\anaconda3\\envs\\d2l\\lib\\site-packages\\d2l\\torch.py\n\u001b[1;31mType:\u001b[0m      function\n"
    }
   ],
   "source": [
    "d2l.load_array??"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [],
   "source": [
    "batch_size, n_train = 16, 600\n",
    "# Only the first `n_train` examples are used for training\n",
    "train_iter = d2l.load_array((features[:n_train], labels[:n_train]),\n",
    "                            batch_size, is_train=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": "torch.Size([16, 4]) torch.Size([16, 1])\n"
    }
   ],
   "source": [
    "for X, y in train_iter:\n",
    "    print(X.shape,y.shape)\n",
    "    break"
   ]
  },
  {
   "source": [
    "在这里，我们保持架构相当简单：只有一个具有两个完全连接的层，ReLU激活和平方损耗的MLP"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Function for initializing the weights of the network\n",
    "def init_weights(m):\n",
    "    if type(m) == nn.Linear:\n",
    "        torch.nn.init.xavier_uniform_(m.weight)\n",
    "\n",
    "# A simple MLP\n",
    "def get_net():\n",
    "    net = nn.Sequential(nn.Linear(4, 10),\n",
    "                        nn.ReLU(),\n",
    "                        nn.Linear(10, 1))\n",
    "    net.apply(init_weights)\n",
    "    return net\n",
    "\n",
    "# Square loss\n",
    "loss = nn.MSELoss()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": "epoch 1, loss: 0.083669\nepoch 2, loss: 0.063961\nepoch 3, loss: 0.064074\nepoch 4, loss: 0.056966\nepoch 5, loss: 0.056334\n"
    }
   ],
   "source": [
    "def train(net, train_iter, loss, epochs, lr):\n",
    "    trainer = torch.optim.Adam(net.parameters(), lr)\n",
    "    for epoch in range(epochs):\n",
    "        for X, y in train_iter:\n",
    "            trainer.zero_grad()\n",
    "            l = loss(net(X), y)\n",
    "            l.backward()\n",
    "            trainer.step()\n",
    "        print(f'epoch {epoch + 1}, '\n",
    "              f'loss: {d2l.evaluate_loss(net, train_iter, loss):f}')\n",
    "\n",
    "net = get_net()\n",
    "train(net, train_iter, loss, 5, 0.01)"
   ]
  },
  {
   "source": [
    "## 8.1.3。预测\n",
    "由于训练损失很小，我们希望我们的模型能很好地工作。让我们看看这在实践中意味着什么。首先要检查的是模型能够很好地预测在下一个时间步中发生的情况，即一步一步预测"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "tensor([5., 6., 7., 8., 9.])"
     },
     "metadata": {},
     "execution_count": 67
    }
   ],
   "source": [
    "time[tau:][0:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "torch.Size([996, 1])"
     },
     "metadata": {},
     "execution_count": 68
    }
   ],
   "source": [
    "onestep_preds = net(features)\n",
    "onestep_preds.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "tensor([[-0.2699],\n        [-0.2124],\n        [ 0.0025],\n        [ 0.1037],\n        [ 0.1498],\n        [ 0.2127],\n        [ 0.1416],\n        [ 0.2187],\n        [ 0.2404],\n        [ 0.3260]], grad_fn=<SliceBackward>)"
     },
     "metadata": {},
     "execution_count": 72
    }
   ],
   "source": [
    "onestep_preds[0:10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "tensor([[0.],\n        [0.],\n        [0.],\n        [0.]])"
     },
     "metadata": {},
     "execution_count": 73
    }
   ],
   "source": [
    "a = torch.zeros(tau,1)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "torch.Size([1000, 1])"
     },
     "metadata": {},
     "execution_count": 81
    }
   ],
   "source": [
    "onestep_preds = torch.cat((a, onestep_preds),0)\n",
    "onestep_preds.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
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      "needs_background": "light"
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   ],
   "source": [
    "\n",
    "d2l.plot(time, [d2l.numpy(x), d2l.numpy(onestep_preds)], 'time',\n",
    "         'x', legend=['data', '1-step preds'], xlim=[1, 1000],\n",
    "         figsize=(6, 3))"
   ]
  },
  {
   "source": [
    "就像我们期望的那样，一步一步的预测看起来不错。即使超出 604 (`n_train + tau`) 个观测值，这些预测仍然看起来值得信赖。但是，这仅有一个小问题：如果我们仅在时间步骤604之前观察序列数据，我们就无法希望收到所有未来所有提前一步预测的输入。相反，我们需要一次向前迈出一步：\n",
    "\n",
    "$$\n",
    "\\hat{x}_{605} = f(x_{601}, x_{602}, x_{603}, x_{604}), \\\\\n",
    "\\hat{x}_{606} = f(x_{602}, x_{603}, x_{604}, \\hat{x}_{605}), \\\\\n",
    "\\hat{x}_{607} = f(x_{603}, x_{604}, \\hat{x}_{605}, \\hat{x}_{606}),\\\\\n",
    "\\hat{x}_{608} = f(x_{604}, \\hat{x}_{605}, \\hat{x}_{606}, \\hat{x}_{607}),\\\\\n",
    "\\hat{x}_{609} = f(\\hat{x}_{605}, \\hat{x}_{606}, \\hat{x}_{607}, \\hat{x}_{608}),\\\\\n",
    "\\ldots\n",
    "$$\n",
    "\n",
    "通常，对于观察到的序列 $x_t$, 其预测输出 $\\hat{x}_{t+k}$ 在时间步 $t+k$ 称为 k提前预测。由于我们观察到 $x_{604}$,  它的  k 提前预测是 $\\hat{x}_{604+k}$.\n",
    "换句话说，我们将不得不使用自己的预测进行多步提前预测。让我们看看这进展如何"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
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   ],
   "source": [
    "multistep_preds = torch.zeros(T)\n",
    "multistep_preds[:n_train + tau] = x[:n_train + tau]\n",
    "for i in range(n_train + tau, T):\n",
    "    multistep_preds[i] = d2l.reshape(net(\n",
    "        multistep_preds[i - tau:i].reshape(1, -1)), 1)\n",
    "\n",
    "d2l.plot([time, time[tau:], time[n_train + tau:]],\n",
    "         [d2l.numpy(x), d2l.numpy(onestep_preds),\n",
    "          d2l.numpy(multistep_preds[n_train + tau:])], 'time',\n",
    "         'x', legend=['data', '1-step preds', 'multistep preds'],\n",
    "         xlim=[1, 1000], figsize=(6, 3))"
   ]
  },
  {
   "source": [
    "如以上示例所示，这是一个惊人的失败。在经过几个预测步骤后，这些预测很快就会衰减到一个常数。为什么算法工作这么差？这最终归因于错误累积的事实。让我们说，在第一步之后，我们有一些错误 ϵ1=ϵ¯ 。现在，第2步的输入受到 ϵ1 ，因此我们遇到了一些错误，其顺序为  ϵ2=ϵ¯+cϵ1  持续不断  c ， 等等。该错误可能与真实的观察结果相去甚远。这是普遍现象。例如，未来24小时的天气预报往往非常准确，但除此之外，准确度会迅速下降。在本章及以后的章节中，我们将讨论改善此问题的方法。\n",
    "\n",
    "让我们仔细看一下  k 通过计算整个序列的预测来提前预测  k=1,4,16,64 ."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
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   ],
   "source": [
    "max_steps = 64\n",
    "\n",
    "features = torch.zeros((T - tau - max_steps + 1, tau + max_steps))\n",
    "# Column `i` (`i` < `tau`) are observations from `x` for time steps from\n",
    "# `i + 1` to `i + T - tau - max_steps + 1`\n",
    "for i in range(tau):\n",
    "    features[:, i] = x[i:i + T - tau - max_steps + 1].T\n",
    "\n",
    "# Column `i` (`i` >= `tau`) are the (`i - tau + 1`)-step-ahead predictions for\n",
    "# time steps from `i + 1` to `i + T - tau - max_steps + 1`\n",
    "for i in range(tau, tau + max_steps):\n",
    "    features[:, i] = d2l.reshape(net(features[:, i - tau:i]), -1)\n",
    "\n",
    "steps = (1, 4, 16, 64)\n",
    "d2l.plot([time[tau + i - 1: T - max_steps + i] for i in steps],\n",
    "         [d2l.numpy(features[:, tau + i - 1]) for i in steps], 'time', 'x',\n",
    "         legend=[f'{i}-step preds' for i in steps], xlim=[5, 1000],\n",
    "         figsize=(6, 3))"
   ]
  },
  {
   "source": [
    "这清楚地说明了我们在尝试进一步预测未来时预测质量的变化。尽管提前4步的预测看起来仍然不错，但超出此范围的任何预测几乎都是无用的。"
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